1. Learning Resource Services
CLAST Review Algebra
Learning Resource Services

2. Real Numbers Real numbers are: Rational numbers Irrational numbers
either terminating (1.0, -5.2) or
repeating decimals ( )
Irrational numbers
non-terminating and non-repeating decimals (, )
All the rules that apply to rational numbers in
arithmetic also apply to irrational numbers.
CLAST Algebra

3. Addition of Irrational Numbers
To add or subtract irrational numbers, first simplify each irrational term and then combine like terms.
Example: 2   all terms are simplified
combine like terms 2 -  + 4 = 5 + 10
Example:
Simplify terms
Combine like terms
CLAST Algebra

4. Multiplication of Irrational Numbers
Example:
Combine radicands under one radical sign
Factor
Simplify
CLAST Algebra

5. Division of Irrational Numbers
Example:
Combine radicands under one radical sign
Factor
Simplify
CLAST Algebra

6. Rationalize the Denominator
Do not leave a radical in the denominator
Example:
Multiply the radical denominator by itself to form a perfect square. Multiply the numerator by the same term, to keep the value of the fraction (this is actually multiplying by 1)
Simplify
CLAST Algebra

7. apply the Order of Operations Agreement
By convention, there is a specific order of operations.
This mnemonic will help you to remember the order:
Please Excuse My Dear Aunt Sally
Parentheses solve the terms inside parentheses first
Exponents evaluate exponents
Multiply and/or Divide from left to right
Add and/or Subtract from left to right
CLAST Algebra

8. use Properties of Operations
Associative (grouping)
a + (b + c) = (a + b) + c
a(bc) = (ab)c
Commutative (order)
a + b = b + a
ab = ba
Distributive
a(b + c) = ab + ac
Inverse
a + (-a) = 0
a ( ) = 1 if a 0
Identity
a + 0 = a
CLAST Algebra

9. use Scientific Notation
Scientific notation always appears as a factor
(one non-zero integer before the decimal portion)
multiplied by a power of 10
Decimal Form Scientific Notation x x , x 104
CLAST Algebra

10. Determine if a Number is a Solution to an Inequality
Is x = 5 a solution to the inequality (x - 5)(x + 4)  0 ?
Substitute 5 for x and simplify each side
(x - 5)(x + 4)  0
(5 - 5)(5 + 4) 0
(0) (9)
zero is equal to zero
x = 5 is a solution
CLAST Algebra

11. Properties of Equality and Inequality
i. If a = b, then a + c = b + c
ii. If a > b, then a + c > b + c
iii. If a = b, then ac = bc
iv. For c > 0, if a > b, then ac > bc
v. For c < 0, if a > b, then ac < bc
vi. If a > b and b > c, then a > c
CLAST Algebra

12. Use Properties to Identify Equivalent Equations and Inequalities
Example: 2 x + 1 = 11
use property i x (-1) = 11 + (-1)
2 x = 10
Example: , given y > 0
use property iv.
CLAST Algebra

13. Solve Linear Equations and Inequalities
To solve an equation or inequality, find its solution.
A solution is a value which makes the equation or
inequality true when the value is substituted for the
variable.
Example: Determine whether x = -1 is a solution of the equation x + 5 = 4.
= Substitute -1 for x.
4 = A true statement
Therefore, x = -1 is a solution
CLAST Algebra

14. Steps to Find the Solution for an Equation
Remove grouping symbols (parentheses) using the distributive property.
Use properties to move all variable terms to one side of the equation.
Use properties to move all non-variable terms to the other side of the equation.
Simplify each side of the equation.
Multiply both sides of the equation by the reciprocal of the numerical coefficient of the variable term. A number multiplied by its reciprocal equals 1:
CLAST Algebra

15. Use Algebraic Formulas
An algebraic formula establishes a relationship between two or more variables.
Example: I = P • R • T
Interest = Principle • Rate • Time
CLAST Algebra

16. Steps to Evaluating Formulas
Identify any given values
Substitute given values into the formula in place of the
variables
Solve the formula for the remaining variable
Label your answer with the appropriate unit
CLAST Algebra

17. Functions
A function establishes a special kind of relationship between two sets of variables. One set is called the independent variable (usually the x values). All of the possible x values are called the domain of the function. The other set is called the dependent variable (usually the y values). All of the possible y values are called the range of the function.
A function relates domain to range
Each value of the domain corresponds to one and only one value of the range. That is, each value assumed by x determines one and only one value of y.
CLAST Algebra

18. Examples of Functions A set of ordered pairs (x and y values)
{(1, 2), (2, 4), (3, 6)}
An equation
y = 2x
An equation written in functional notation
f(x) = 2x
CLAST Algebra

19. Visual Examples of Functions
Remember, a function is defined so that each value of the domain corresponds to one and only one value of the range. One method to visually determine whether a graph represents a function is to apply the “vertical line” test. If a vertical line passed over the graph touches more than one point at a time, the graph does not represent a function. .
not a function .
function
CLAST Algebra

20. Find Values of Functions
Functional notation states the value which is to be substituted for x into an equation in order to determine the corresponding value for y:
Example: Given f(x) = 2x - 3, find f(2)
Substitute the value 2 for x in the equation
f(2) = 2(2) - 3
= 4 - 3
= 1
CLAST Algebra

21. Find Factors of Quadratic Expressions
A quadratic expression contains one term in which the variable is squared.
Example: 2x2 + 5x + 2
The factors of a quadratic expression are two linear expressions which multiply together to form the original quadratic expression.
(2x + 1)(x + 2) = 2x2 + 4x + 1x + 2
= 2x2 + 5x + 2
Outside terms
Last terms
First terms
Inside terms
CLAST Algebra

22. Find Factors of Quadratic Expressions page 2
Example: (3x - 2)(x + 4) = 3x2 + 12x - 2x - 8
= 3x2 + 10x - 8
The two linear factors are determined by the FOIL technique:
The product of the First term of each factor is the first term of the quadratic
The products of the Outside terms and the Inside terms of the factors, added together, is the middle term of the quadratic
The product of the Last term of each factor is the last term of the quadratic
Outside terms
Last terms
First terms
Inside terms
CLAST Algebra

23. The FOIL Technique
Example: Determine the linear factors of the quadratic expression 4x2 - 9x + 2
Step 1: Determine all possible pairs of factors of the First term x and x 2x and 2x
-4x and -x x and -2x
Step 2: Determine all possible pairs of factors of the Last term and and -1
CLAST Algebra

24. The FOIL Technique page 2
4x2 - 9x + 2
Step 3: Systematically test one pair of factors from step 1 with one pair of factors from step 2 until the middle term of the quadratic is formed as the sum of the products of the Outside and Inside terms.
(4x + 2)(x + 1) 4x + 2x = 6x No
(4x + 1)(x + 2) 8x + 1x = 9x No
(4x - 2)(x - 1) x - 2x = -6x No
(-4x - 2)(-x - 1) x + 2x = 6x No
(4x - 1)(x - 2) x - 1x = -9x Yes
through trial and error, 4x2 - 9x + 2 = (4x - 1)(x - 2)
Outside +
Inside
CLAST Algebra

25. Find Solutions to Quadratic Equations
The solution to a quadratic equation of the form ax2 + bx + c = 0 can be found by
factoring
set each factor equal to zero, then solve
each of the resulting linear equations
substituting the values for a, b, and c into the
quadratic formula:
CLAST Algebra

26. A System of Two Linear Equations
A system consists of two or more linear equations, each of the form ax + by = c. Each equation has an infinite number of ordered pair (x, y) solutions.
A system represents two lines on the coordinate plane. The solution of the system can be
one ordered pair (the two lines intersect)
no ordered pairs (the two lines do not intersect), or
an infinite number of ordered pairs (the two lines are the same)
CLAST Algebra

27. Solve a System of Two Linear Equations in Two Unknowns: the Addition Method
Write each equation in standard form. Determine which coefficients of the x and y terms will be the easiest to make into opposite values.
If necessary, multiply one or both of the equations by a constant to make the chosen set of coefficients opposites.
Add these two new equations. The result is a single equation with one variable. Solve for this variable.
Substitute this value into either of the original equations and solve for the other variable. Check your answer.
CLAST Algebra

28. Example of the Addition Method
Solve the system of equations x - y = 4
3x - 2y = 2
Step 1: The equations are already written in standard form. The y terms will be easier to make opposites.
Step 2: Multiply the first equation by -2
Step 3: The new system of equations is
-4x + 2y = -8
3x - 2y = 2
CLAST Algebra

29. Addition Method page 2 Add the two equations -4x + 2y = -8
Multiply this equation by -1 to determine that x = 6.
Step 4: Substitute x = 6 into the original form of the first equation to solve for y.
2(6) - y = The solution is the ordered pair (6, 8)
y = Check in first equation: 2(6) - 8 = 4
- y = = 4 
y = Check in second equation: 3(6) - 2(8) = 2
= 2 
CLAST Algebra

30. Identify Specified Regions of the Coordinate Plane
A coordinate plane is where ordered pairs are graphed in relation to a horizontal axis (x) and a vertical axis (y). A linear inequality, when graphed, produces not just one solution, but a region of solutions.
Example: 2x - y < 6
Step 1: Sketch the boundary line 2x - y = 6
x y y . x
boundary line
< dashed < solid
> dashed > solid .
CLAST Algebra

31. Regions of the Coordinate Plane page 2
Step 2: Shade the region that contains the solutions
Choose a test point not on the boundary line and substitute this point into the original inequality.
Let the test point be (0, 0)
2x - y < 6
2(0) - 0 < 6
0 < 6 a true statement
If true, shade the region that contains the test point
If false, shade the region that does not contain the test point y . x • .
CLAST Algebra

32. Problems Involving the Structure and Logic of Algebra
An algebraic relationship between two or more quantities may sometimes be stated as a word problem. The words need to be translated into math symbols so that a solution can be determined.
Choose a variable to represent one quantity
Express all other quantities in terms of this variable
Express the relationship symbolically as an equation or inequality
CLAST Algebra

33. Solve Problems Involving the Structure and Logic of Algebra
Example: Write the equation needed to find a number if that number, increased by five times that number, is 48.
Let n represent the number
5n represents five times the number
+ represents “increased by”
= represents “is”
The equation needed is n + 5n = 48
CLAST Algebra

34. Identify Statements of Proportionality and Variation
Two quantities, represented by the variables x and y, can be related in many ways. Two special relationships are that
x and y vary directly: x produces y
Direct Variation equation y = kx or Ratio
x and y vary inversely: x produces y
Inverse Variation equation xy = k or Ratio
k is the constant of proportionality
CLAST Algebra

35. Example of Direct Variation
When traveling, distance equals rate times time, or d = rt
Distance increases if rate or time increases
Distance decreases if rate or time decreases
Example:
If r = 10 miles/hour
and t = 1 hour,
then
d = 10m/hr x 1 hr
= 10 miles
If r increases to 20 miles/hour
and t remains 1 hour,
then
d = 20m/hr x 1 hr
= 20 miles (an increase)
CLAST Algebra

36. Example of Inverse Variation
When traveling, distance equals rate times time, or d = rt .
This formula can be solved for r, as r =
If distance remains the same, rate will change inversely to time
Example:
If d = 10 miles
and t = 1 hour,
then
r = 10 miles/1 hour
= 10 miles/hour
If d remains 10 miles
and t increases to 2 hours,
then
r = 10 miles/2 hours
= 5 miles/hour
CLAST Algebra

37. Algebraic Word Problems with Variables
The algebraic solution to a word problem includes the following steps:
Read the problem carefully
Organize the problem on paper
Substitute values into a given formula. If no formula is
given, identify the relationship between the various
quantities and express that relationship as an equation.
Solve for the unknown variable
Check the solution
CLAST Algebra